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The Klausmeier model of arid vegetation patterns

When you approach the Sahara desert from the south, rainfall is
in increasingly lower supply. You would expect the vegetation to
gradually become sparser and sparser, until nothing else remains
than desert sand.

On the edge of the desert, however, something remarkable occurs.
The bushland vegetation starts to develop strikingly regular patterns,
in the form of stripes, dots, gaps, or over labyrinth shapes.

To understand how these patterns form, a young Chris Klausmeier made
a very simple model, that deals with how vegetation interactions with
rainwater. Rainwater does not infiltrate well on the clayey soils where
these patterns are typically found. However, where plants grow, they
open up the soil, allowing rainfall to infiltrate, stimulating
vegetation growth, leading to even better infiltration. In other words,
where more plants grow, there is more water for the plants to consume.

Chris made a very simple mathematical model to capture this feedback
processes. This model described the local dynamics of water
w and plants n:

∂w / ∂t = a - w - wn2 - v ∂w / ∂x

∂n / ∂t = wn2 - mn
+ ∂2n / ∂x2
+ ∂2n / ∂y2

Here, rainfall a is the source for water, which is either taken up by
plants, evaporates, or flows downslope. Plant uptake of water increases
disproportionally with plant density because of the stimulation of
water infiltration by the plants, which is captured by the term
wn2.
Plant growth is linearly dependent on water update, and plants face
losses due to senescence with a rate m. Water movement on hill slopes
as modelled by the advection term v ∂w / ∂x,
where v is the water flow rate. Finally, vegetation spread is modelled
as a diffusive process. On flat surfaces, where water flow downhill is
negligible, water flow is similarly approximated as a diffusive process.

Despite of its simplicity, this model is capable of explaining the
majority of spatial patterns that are observed in arid lands.

Try the above java applet to get aquainted with the model. The applet
actually runs the following model:

∂w / ∂t = a - w - wn2 - v∇w + d∆n

φ ∂n / ∂t = wn2 - mn + ∆n

where ∇ and ∆ are the gradient and laplacian operators, respectively,
and d represents water dispersion on flat soil. φ respresent an
acceleration parameter that bridges the time gap between water
movement and plant growth.